Details of Grant 

EPSRC Reference:	EP/Y001478/1
Title:	Large-N limit of horizontal Brownian motions on Lie groups
Principal Investigator:	Habermann, Dr K
Other Investigators:
Researcher Co-Investigators:
Project Partners:	University of Aveiro
Department:	Statistics
Organisation:	University of Warwick
Scheme:	Standard Research - NR1
Starts:	01 October 2023	Ends:	30 September 2024	Value (£):	56,725
EPSRC Research Topic Classifications:	Mathematical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:	Panel Date Panel Name Outcome 17 May 2023 ECR International Collaboration Grants Panel 1 Announced
Summary on Grant Application Form
This collaborative project aims to explore large-N limits of natural horizontal Brownian motions on Lie groups of N x N matrices. It lies at the intersection of probability theory, differential geometry and group theory, and particularly combines the study of stochastic processes on Lie groups and the study of sub-Riemannian geometries in a novel way. Sub-Riemannian geometries model systems with constraints but set up such that the system moves over all parts of the phase space, that is, the constraints are flexible enough that any two points in the space can be connected by a curve satisfying the constraints. These type of geometries naturally appear in all sciences, ranging from constrained physical systems over motion planning in robotics to modelling the first layer of the visual cortex of the brain. For instance, the position of a vehicle in a field can be described by specifying the coordinates of its centre and the angle of rotation with respect to a reference line. In this three-dimensional parameter space, it is not possible to perform motions which correspond to a movement perpendicular to the direction of the wheels of the vehicle. However, by choosing suitable maneuvers it is still possible to reach any target position. Large-N limits of Brownian motions on Lie groups of N x N matrices have been actively studied. The analysis employs tools from free probability and the results have implications in random matrix theory. In these works, the Lie groups in considerations are equipped with a canonical Riemannian structure. We plan to now tackle the natural question of what can be said about the large-N limits of horizontal Brownian motions on Lie groups of N x N matrices where, instead of using a Riemannian structure, the Lie groups are equipped with canonical sub-Riemannian structures.
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Further Information:
Organisation Website:	http://www.warwick.ac.uk